Oscillations: Complete A Level H2 Physics Cheatsheet
A Level Physics (H2), Chapter 9 · 9478 Topic 9 · Read time: ~10 minutes
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Key Formulas
Free oscillations and SHM
A free oscillation has no energy gained from or lost to the surroundings, so it oscillates at its natural frequency with constant amplitude. Examples: a pendulum or a mass on a spring, ideally.
Simple harmonic motion: acceleration is directly proportional to displacement from equilibrium and always directed towards it: a = −ω²x. The negative sign is what makes the motion return.
Key terms: amplitude x₀, period T, frequency f, angular frequency ω = 2π/T, and phase difference, which compares two oscillations in radians or as a fraction of a cycle.
Trap: At maximum displacement the speed is zero but the acceleration is maximum, pointing back towards equilibrium. At equilibrium the speed is maximum and the acceleration is zero.
Displacement, velocity and acceleration
If x = x₀ sin ωt, then v = v₀ cos ωt and a = −ω²x₀ sin ωt.
Velocity leads displacement by π/2. Acceleration is in antiphase (π) with displacement.
On an a-x graph, SHM gives a straight line through the origin with gradient −ω².
Trick: To find speed at a given displacement, use v = ω√(x₀² − x²) directly. It avoids needing the time or the starting phase.
Energy in SHM
Energy swaps continuously between kinetic and potential stores. Kinetic is maximum at equilibrium; potential is maximum at the extremes.
Total energy E = ½mω²x₀² stays constant if undamped. Kinetic energy at displacement x is ½mω²(x₀² − x²).
Both Ek and Ep go through two maxima per oscillation, so their graphs against time have twice the frequency of the motion.
Damping and resonance
Damping removes energy, so amplitude decreases. Light damping: many oscillations, slowly decaying. Critical damping: returns to equilibrium in the shortest time, without oscillating, as in a car suspension. Heavy damping: no oscillation, but a slower return.
In a forced oscillation, a driver makes the system oscillate at the driving frequency. Amplitude is greatest at resonance, when the driving frequency is close to or at the natural frequency.
More damping lowers the resonance peak, makes it broader, and shifts it to a slightly lower frequency.
Resonance is useful in MRI scanners, microwave ovens and tuning a radio; it must be avoided in bridges and buildings, where large amplitudes cause damage.
Worked Example
A 0.20 kg object oscillates with SHM, with amplitude 0.050 m and period 2.0 s.
(a) Calculate ω and the maximum speed. [2]
ω = 2π/T = 3.14 rad s⁻¹
v₀ = ωx₀ = 3.14 × 0.050 = 0.157 m s⁻¹
(b) Calculate the speed when the displacement is 0.030 m. [2]
v = ω√(x₀² − x²) = 3.14 × √(0.0025 − 0.0009)
= 3.14 × 0.040 = 0.126 m s⁻¹
(c) Calculate the total energy and the kinetic energy at x = 0.030 m. [2]
E = ½mω²x₀² = 0.5 × 0.20 × 9.87 × 0.0025 = 2.47 × 10⁻³ J
Ek = ½mω²(x₀² − x²) = 0.5 × 0.20 × 9.87 × 0.0016 = 1.58 × 10⁻³ J
Why this question is a good test of the topic: every part uses x₀ and ω, without ever needing the time. That's the fast route examiners expect in SHM energy questions.
Frequently Asked Questions
What is the defining equation of SHM?
a = −ω²x. Acceleration is proportional to displacement and always directed towards the equilibrium position.
What is the phase relationship between x, v and a in SHM?
Velocity leads displacement by π/2. Acceleration is in antiphase with displacement.
Why is critical damping used in car suspensions?
It returns the car to equilibrium in the shortest time without bouncing.
What is resonance?
Maximum amplitude of a forced oscillation when the driving frequency is at or close to the natural frequency of the system.
How does damping affect resonance?
It reduces the peak amplitude, broadens the peak and shifts it slightly to a lower frequency.
SHM graphs still confusing?
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